10/21/2026, 03:37 PM
Just a brief update: the paper mentioned in my previous post, "A Compact Notation for Peculiar Properties Characterizing Integer Tetration", has just been published in its final (peer-reviewed) version in Notes on Number Theory and Discrete Mathematics, 32(3) (2026), 477–496.
In particular, Appendix B establishes that squarefree radices are precisely those for which the congruence speed of integer tetration eventually stabilizes to a positive constant for every integer base greater than \(1\) not divisible by the radical of the radix.
The paper also investigates the behavior for non-squarefree radices and proves the following multiplicative lower-bound property, at least for the decimal numeral system:
\(V(x \cdot y) \geq \min\{V(x),V(y)\}\),
for all integers \(x,y>1\) such that \(10 \nmid x \cdot y \).
In particular, Appendix B establishes that squarefree radices are precisely those for which the congruence speed of integer tetration eventually stabilizes to a positive constant for every integer base greater than \(1\) not divisible by the radical of the radix.
The paper also investigates the behavior for non-squarefree radices and proves the following multiplicative lower-bound property, at least for the decimal numeral system:
\(V(x \cdot y) \geq \min\{V(x),V(y)\}\),
for all integers \(x,y>1\) such that \(10 \nmid x \cdot y \).
Let \(G(n)\) be a generic reverse-concatenated sequence. If \(G(1) \notin \{2, 3, 7\}\), then \(^{G(n)}G(n) \pmod {10^d}≡^{G({n+1})}G({n+1}) \pmod {10^d}\), \(\forall n \in \mathbb{N}-\{0\}\)
("La strana coda della serie n^n^...^n", p. 60).
("La strana coda della serie n^n^...^n", p. 60).
